Pricing as a risky choice: Uncertainty and survival in a monopoly market

Roy (Safety First and the Holding of Assets, 1952) argues that decisions under uncertainty motivate firms to avoid bankruptcy. In this paper the authors ask about the behaviour of a monopolist who pre-commits to price when she has only probabilistic knowledge about demand. They argue that pricing in order to maximise the likelihood of survival explains anomalies such as inelastic pricing, why the firm takes on more risk as gains become less likely, and asymmetric responses to demand and cost changes. When demand is a linear demand, the monopolist’s response to an increase in the marginal cost is similar to the response when mark-up pricing is used. That is, there is a oneto-one relationship between an increase of the marginal cost and an increase in price. JEL D42 L12 L21


Introduction
Following Sandmo (1971), one of the shortcomings of assuming that firms maximise expected profits when they make decisions under uncertainty, is that they ignore the consequences of losses, including the possibility of bankruptcy. For example, consider a firm that has two possible strategies. Suppose the first strategy offers marginally higher expected profit than the second strategy. If pure profit maximisation guides the firm's decision, it chooses the first strategy even if this strategy involves a strictly higher likelihood of bankruptcy. As pointed out by Roy (1952), it makes sense that firms maximise the expected profit or expected utility as a function of profit if performances in good states of nature finance bad outcomes. In general, though, investors and firms do not take part in a lottery that is repeated a large number of times. Realistically, a poor performance does not always trigger another chance to make up for a current dread event.
One of the alternatives to profit maximisation, with respect to decision-making under uncertainty, is that the primary objective of most investors and firms is to avoid results that close down a business (Roy, 1952).
Under the safety-first principle, firms minimise the probability of bankruptcy or profits falling below some threshold that means the end of further business. 1 It can be argued that the safety-first principle puts too much emphasis on risk relative to income, and Telser's (1955) subsequent extension of Roy's decision principle includes the possibility that firms maximise expected profits subject to some upper limit on the probability of bankruptcy. A second alternative is that firms maximise expected profits less the weighted standard deviation of profit (Turnovsky, 1968). In this paper, we apply the safety-first criterion to a 1 From a practical perspective notice that the most widely used measures in risk management are varieties of the safety-first principle (see Chiu , Wong and Li, 2012). Also, executive directors have incentive to follow the safety-first principle because, as shown by Hilger et al. (2013), financial distress significantly increases the likelihood of executive dismissal. Schmidt (1997) analyses how the threat of liquidation is used in managerial incentive schemes. monopolistic firm and ask to what extent the resulting price behaviour is in agreement with some of the observation anomalies of firms' pricing practices.
There are plenty of results on the topic of the non-expected utility-maximising behaviour of individual decision makers and, with respect to firms, this literature suggests that it is relevant to look for motives that supplement profit maximisation as an explanation for pricing behaviour. With respect to setting prices, Cyert and March (1963) suggest that monopolies apply routine decisions in the form of cost-plus pricing.
Empirical studies of firms' pricing behaviour by Koutsoyiannis (1984) and van Dalen and Thurik (1998) dismiss the assumption that they maximise profit. More recently, Fabiani et al. (2007) show that a significant number of firms in Europe apply mark-up pricing. Secondly, empirical findings on the pricing of performance goods (Marburger,1997;Forrest, Simmons and Feehan, 2002;Krautman and Berri, 2007) suggest that firms engage in inelastic pricing. This is also confirmed for the Swedish Tobacco Monopoly's sale prices (Asplund, 2007). Thirdly, Kahneman and Tversky (1979) suggest that risk becomes more acceptable as gains become less likely. Finally, Kahneman, Knetsch and Thaler (1986) argue that concerns for fairness can explain why price responds more strongly to cost changes than to demand changes. Of course, it is not likely that all firms follow a safety-first strategy, but we find it of some interest that the pricing behaviour of a price-setting monopolist who is motivated by the safety-first principle agrees, to a certain extent, with the abovementioned observations. The model we apply is that of a price-setting monopolist in a market with stochastic demand under the assumption that the monopolist sets a price that determines her sale. 2 Because the price decision occurs before the actual value of demand becomes known, the monopolist runs the risk of realising profits below 2 Allen and Hellwig (1986) notice that price setting describes an essential feature of many markets. a minimum acceptable return. Our focus is (primarily) the case where the minimum acceptable return is zero and, therefore, we examine the pricing behaviour that minimises the likelihood of negative profits. But more generally, the safety-first principle takes into consideration poor performances that correspond to the tail part of a profit's distribution. 3 Thus, applying the safety-first criterion to monopolistic pricing, we think of pricing as a risky choice, essentially looking at the situation where the firm cannot diversify away the risk of ruin by drawing on gains in future time-periods, by engaging in multiproduct activities or in physically separate markets, etc., or through insurance. When the monopolistic firm's decision leaves profit's variance unaffected, this means that the firm maximises expected profit.
In comparison to standard results, there are several consequences when the safety-first principle guides the monopolist's pricing decision. First, when a higher price means higher variance of profit, the price under the safety-first criterion falls short of the price that maximises expected profit. This conclusion is general in the sense that the result-that the optimum price under the safety-first criterion is below the profit maximising price-applies when the objective is to maximise a utility function defined over expected profit and the risk of bankruptcy. Also, as long as the profit function's variance is increasing in price, our results apply irrespective of the threshold being negative (when the firm has access to credit) or positive (which seems relevant when the manager is fired as a consequence of profits below a certain threshold).
Second, with respect to comparative statics, when demand increases by rotation, the optimum price under the safety-first principle goes down while, under a linear demand curve, a shift in willingness to pay does not affect the optimum price. In contrast, for a monopolist who maximises expected profit, comparative 3 When the gain is normally distributed, the safety-first principle implies that the objective is to maximise the difference between expected return and a threshold return relative to the standard deviation. In symbols, the objective function is where ̅ is the minimum gain, ( ) the expected gain, and √ ( ) the standard deviation. When the minimum acceptable gain is zero the firm maximises ( ) √ ( ) ⁄ .
static results suggest that the price is unchanged and increasing. With respect to cost-changes, a higher fixed cost increases the optimum price under the safety-first criterion while the price is unrelated to a fixed cost when firms maximise expected profit. Also, when the firm applies the safety-first principle and demand is convex, the pass-through for an increase of the unit cost exceeds 1 and, as we show, this can result in profit-increasing taxes.
Previous work on monopolistic firms' behaviour under demand uncertainty applies the assumption that the monopolist maximises the utility of expected profit. Thus, Baron (1971), Leland (1972), and more recently Hau (2004), analyse the monopolist's optimal response to risk under the assumption that the firm survives under any realisation of demand. Kimball (1989), who assumes that marginal costs are convex, shows that a monopolist who commits to price before observing demand will charge a higher price as uncertainty increases. This result is based on the observation that marginal production cost increases in all states for a mean-preserving spread, and, thus, ignores the chance that the firm ends up in a situation of financial distress. Although Sattinger (2013) applies the safety-first principle to savings decisions, and Chiu at al.
(2012) use it to discuss portfolio choice, problems relating to economic survival seem to be ignored in recent discussions of firms' behaviour under risk. Our discussion is probably most closely related to Day, Aigner and Smith (1971) and Arzac (1976). Discussing safety margins versus expected profit maximisation for a quantity-setting monopolist, Day et al. (1971) show that output falls short of the quantity that maximises expected profits when avoidance of ruin matters. This conclusion stands when the monopolist follows a pure safety-first criterion and when she uses some other rules for aggregating expected profit and safety margin into one performance measure. Later, Arzac (1976) affirms these results without making any specific assumption about the shock, and, under the pure safety-first criterion, shows some comparative static results: an increase in willingness to pay as well as an increase of a unit tax rate leaves the optimum value of output unchanged. Our findings reverse existing conclusions on the implications of the safety-first principle, as we show that a price-setting monopolist hedges against risk by lowering the price relative to the one that maximises expected profit. Also, as we noted, a change of the constant marginal cost, inclusive of an increase of the unit tax rate, tends to drive up the optimum price.
The paper proceeds as follows. Section 2 models how a monopolist prices following the safety-first principle. Section 3 explores comparative statics and the possibility of profit-increasing excise taxes, and illustrates the asymmetries in adjusting for cost and demand changes. Section 4 discusses the possibility of pricing in the inelastic part of the demand curve. Section 5 concludes.

Price decisions and survival
We consider a monopolistic firm that sets a price before the exact value of demand is observed. More precisely, the firm is placed in a market where the relationship between sale, called ( , ), and the price the monopolist charges, called , is stochastic according to: Here is a continuous stochastic variable with an expected value of zero and a variance of 2 . Although the monopolist knows the distributional properties of , she is ignorant about the exact realisation of the stochastic variable when she sets price. For a given price, the monopolist's prediction of sale is equally accurate across the realisations of the stochastic innovation because of the assumption that uncertainty is additive. 4 For simplicity, we focus on a production technology of the type where there is a fixed cost, called , and constant marginal cost called . That is, the firm's production cost is: Realised profit, called , is a stochastic variable given by: Of course, when the monopolist commits to a price, her beliefs about the expected value of profit and the variance of that profit are both functions of the price. More precisely, the expected value of profit and variance are given by the following equations (4) and (5), respectively.
It is easy to see from equations (4) and (5) that actual profits can be negative even when expected profits are positive. For example, suppose that maximises expected profit, and assume that ̅( ) is positive.
Because the price co-determines the distribution of profit, the monopolist's choice of price affects the chance of financial distress. By setting the price, the monopolist decides on the mean of profits, ̅( ), and the standard variation of profits, ( − ) . Supposing that profits are normally distributed, we can write actual profit as ( ) = ̅( ) + ( − ), and the probability that profits are negative equals the Under this assumption about the distribution of profits, the firm minimises the likelihood of negative profits by maximising ( ( − )) −1 ̅( ). In general, when profits follow some other distribution than the normal distribution, Tchebycheff's inequality implies that the firm minimises the value of an upper bound on the likelihood of negative profits by maximising ( ( − )) −1 ̅( ). Thus, when the firm follows the safety-first principle, the relevant optimisation problem is: 5 under the assumptions that the second-order condition is satisfied and when we assume that expected profits are non-negative for < ≤ ≤ ℎ , where ̅( ) = ̅( ℎ ) = 0. The restriction that > follows because the monopolist produces with an average loss equal to the fixed cost, should she use marginal cost pricing. The arguments leading to equation (6) show that the safety-first principle implies that the monopolist maximises expected profits when her decision on price leaves variance unchanged. The optimisation problem is alternatively written as: Equation (7) implies that the monopolist decides on a price, called * , which maximises the difference between expected sales, ̅ ( , ) = ( ), and (expected) zero-profit sale. The latter term, the price that implies a zero-profit sale, is critical in the sense that setting the price higher produces, on average, negative profits. Therefore, the price that, on average, gives a zero-profit sale is the upper bound on the price the monopolist will set. Maximising the distance between expected sale and the zero-profit sale, the firm maximises the safety margin, i.e., sets a price that minimises the probability of revenue falling short of costs inclusive of the fixed cost. All in all, the optimum price satisfies: We can compare the bankruptcy-minimising price with the price that maximises expected profit. To do so, monopolist engages in production if average profit is positive, i.e., ( * ) − ( * − ) −1 > 0, we have * < . That is, pricing that aims at minimising the likelihood of financial default reduces the price in comparison with the price that maximises expected profit. We summarise this as Proposition 1.
Proposition 1. Under the assumptions that expected profits are positive and variance of profits is increasing in price, the monopolist who uses price to minimise the probability of negative profits, sets a price that is less than the price that maximises expected profit.
To explain the intuition of the result, notice that the pricing policy under the safety-first principle reduces expected profits due to the second-order condition which implies ̅( * ) < ̅( ). However, the reduction in expected profits is compensated for by a gain in terms of reduced risk because ( * ) 2 < ( ) 2 . To see why the firm prefers to reduce risk by pricing aggressively, notice that the price * maximises Ω = (( − ) ) −1 ̅( ). Consider the value of Ω⁄ evaluated at = . Because ̅⁄ = 0 at = , it follows that a marginally lower price around lowers ̅( ) but the decrease is of the order of 2 . In contrast, the increase of ( − ) is of the order of meaning that Ω⁄ is positive around = . In other words, if the monopolist charges the price that maximises expected profit and considers a deviation from this price, the effect on profit is vanishing because the monopolist deviates from a top. However, by reducing the price, she reduces the variance of profit, and this shows why the optimal price is less than the monopoly price. Incidentally, this way of understanding the result points to a generalisation-that the optimal price is less than the monopoly price whenever the profit's variance is increasing in the price, and is and higher than the monopoly price whenever the profit's variance is a decreasing function of the price.
Proposition 2. Whenever the variance of profit is increasing (decreasing) in price, a monopolist who minimises the probability that profit falls short of sets a price that is less (greater) than the price that maximises expected profit.
Assuming that the aim is to maximise expected profit, Kimball (1989) analyses, as we do, how a monopolist sets price when there is only probabilistic knowledge about demand at the time when the price is chosen.
Unlike our approach, Kimball (1989) analyses the situation of multiplicative demand uncertainty. When the monopolist adjusts production to meet demand at the preannounced price, uncertainty drives up the optimum price if production occurs under convex marginal costs. The explanation for this result is that marginal costs increase by a lot, comparatively speaking, when demand increases above its mean relative to the cost decrease that comes with demand decreases below the mean. To hedge against the consequences of extraordinary high costs under high demand, the firm finds it attractive to set a high price.
This argument, based on the observation that a low price forces the firm to substantial sale and costs in states with high demand, ignores the fact that a high price is more likely to lead to actual profits that are negative. Our result shows that, when the monopolistic firm uses price to hedge against the risk of bankruptcy, the optimum price is less than the price that maximises the expected profit, and that a higher price implies higher variability of profit.
Finally, notice that in Proposition 1, a monopolist who follows the safety-first principle, reduces her price relative to the price-maximising expected profit (positive or negative) and generalises to more general preferences than those examined so far. To see this, suppose that the financial restriction is that profits less than are unacceptable. In principle, can take on a negative value if the monopolist can borrow or have other sources of finance. Under the restriction that is the least acceptable profit, the monopolist is active when ( − ) ( ) ≥ + . With a reasoning that parallels the above, the monopolist thus maximises principle. Let us assume that the monopolistic firm's objective is to maximise ( ̅( ), ℎ( )) where utility is increasing in expected profits, 1 = (. ) ̅( ) ⁄ is positive and decreasing in the probability of realising profits below the threshold, and that 2 = (. ) ℎ( ) ⁄ is positive. We have: Proposition 3. When ℎ( ) has a solution, the solution to ( ̅( ), ℎ( )) implies that the price is less than the price that solves ̅( )⁄ = 0 under the assumption that 1 (. , . ) > 0 and 2 (. , . ) > 0 .

Comparative statics
3.1 Demand changes Baldenius and Reichelstein (2000) discuss comparative statics in monopoly and distinguish between uniform changes in consumers' willingness to pay and demand-increases by rotation. Consider uniform changes in willingness to pay. In the standard example of a monopolist in a market with linear demand, the profit-maximising price increases by a rate of half of the size of the shift. When the monopolist who operates under the same linear demand curve follows a safety-first strategy, a uniform change in willingness to pay does not affect the optimum price. The latter conclusion follows from inspection of It is easy to see that * ⁄ = − ( * − ) ′ ( * ) Λ ⁄ where Λ < 0 is the second-order condition corresponding to equation (9). That is, * ⁄ < 0. The reason that the price goes down when demand increases by rotation is that the variance of profit goes up. This makes the incentive for cautiousness stronger and explains why the price goes down. To the contrary, a monopolist who maximises expected profit chooses the price according to ′ ( ) + ( ) − ′ ( ) = 0 and it is straightforward that a demand change by rotation leaves the price unchanged.

Cost changes
It is easy to see from the first-order condition, equation (9), that an increase of the fixed cost results in a higher optimum price. That is, when the monopolist is exposed to higher risk of financial distress through a higher fixed cost, she responds by increasing the price she charges. Kahneman and Tversky (1979) suggest that potential gains and losses account for decisions involving risk. In particular, agents might take on more risk when gains have moderate probabilities. Actually, this is what is happening when the fixed cost increases. Other things equal, this makes the likely occurrence of success go down, and the response of increasing the price implies more risk in the sense that the variance of profit goes up.
It is well known that under linear demand, an increase of the fixed marginal cost increases the price by fifty percent of the cost increase when the monopolist maximises expected profit. In contrast, when optimal pricing follows the safety-first criterion we have: * ⁄ = (( * − ) 3 2 − ′′ ( * )) −1 ( * − ) 3 2 .
Equation (10) shows, in the case of linear demand, that the final price changes one-to-one with a change in the constant marginal cost. When demand is convex, it is evident that the pass-through rate of marginal cost increases is more than a hundred percent. This observation has implications for analysis of tax incidence. Under an excise tax with a tax rate of , the first-order condition in equation (8)  . When ( * ( )) > ½ ( * ( )), the monopolist's expected profit increases with an increase of the tax rate. 7 We have * ( )⁄ = Λ −1 ( ′( * ( ))+( * ( ) − − ) −2 ), where Λ = 2 ′ ( * ( )) + ( * ( ) − − ) −2 ′ * ( * ( )) < 0 is the second-order condition after rewriting by use of the first-order condition.
Proposition 4 summarises the conditions that give rise to profit-increasing taxes. The existence of profitincreasing taxation owes to the fact that the firm, when trying to avoid a dread event, sets a price that is lower than the price that maximises expected profit. Because the tax drives up the price, there is a beneficial profit effect of taxation. When this effect is sufficiently strong, profits go up. Broadly speaking, this happens when the elasticity of ( ) exceeds the elasticity of ′ ( ) to a sufficient degree, or when the graph of ( ) is sufficiently curvy (evaluated at * ( )). This means that the price can increase a lot, comparatively speaking, without strong negative effects on sale.
In a partial model, the standard measure of welfare is the sum of consumer and producer surplus. The positive profit effect of the tax, which comes about because the price goes up, is equivalent to an increase in producer surplus. The price increase is harmful with respect to consumers and it is straightforward to verify, in spite of the positive effect on the monopolist's profit, that the standard conclusion still applies, that the sum of changes in consumer and producer surplus exceeds the tax burden. Following Weyl and Fabinger (2013), the change of the value of expected consumer surplus is: where ( ) = * ( )⁄ is the pass-through. Because ( ) = 0, the change in consumer surplus follows: which implies that ( ⁄ ) = − ( ) ( ( )). The effect on expected profit is ̅⁄ = (( − − ) ′ ( ( )) + ( ( ))) ( ) − ( ( )). Proposition 4 suggests that ̅⁄ is positive for some parameter configurations. In this case, we have ( ) > 1 and it follows that the total burden net of the tax revenue, which is −( − − ) ′ ( ( )) ( ), is positive.

Inelastic Pricing
When the monopolist behaves cautiously and lowers her price, it is possible that the optimum price is in the inelastic region of the demand schedule. To examine this possibility we can rewrite equation (8) as: where ( * ) is the price elasticity of demand at the price * . If the right-hand side in equation (13)  To further explain the occurrence of inelastic pricing, suppose that demand follows = − + . In this situation the first-order condition for * solves for * = + √ −1 . The price that maximises expected revenue is given by = ½ −1 . In turn, the monopolistic firm engages in inelastic pricing whenever the revenue-maximising price exceeds the optimal price, or ≥ 2( + √ ). It is easy to understand this condition by using Figure 1. The slope of the demand function isand the slope of ( − ) −1 is −( − ) −2 . Now, the optimal price is in the inelastic segment of the demand schedule when, evaluated at = , −( − ) −2 ≥ − which is the same thing as ≥ 2( + √ ).
It is immediately clear that a high value of and low values of , and are conducive to inelastic pricing.
Because an increase of drives up the revenue-maximising price while it leaves the optimal price unchanged, a higher means that it is more likely that * falls short of . To state this in a slightly different way, notice that a high value of , other things equal, produces, on average, profits that are significantly higher than the critical profit value (which we set to zero). In this situation, the firm reduces the price in order to reduce the variance. A low value of means that the firm is close to the critical value and it is not beneficial to sacrifice profits for reduced variance. In addition, a decrease of , or drives up the expected profit so that it exceeds the critical value, and the firm then sacrifices some of this profit to obtain reduced variance. Formally, notice that lower values of and drive down the optimal price without affecting the revenue-maximising price, showing why low values of and are associated with inelastic pricing. The effect of a change in is more complex because the optimal and the revenuemaximising prices change. However, showing that the optimal price falls short of the revenue-maximising price as → 0. Figure 1 here.

Discussion
In this paper we ask to what extent the pricing behaviour of a price-setting monopolist guided by the safety-first principle mimics actual pricing practices. In the introduction we listed some of the observations that have been made with respect to pricing. Firstly, the case appears that many firms diverge from the pure profit-maximising behaviour because they use mark-up pricing. When a monopolist applies the safety-first principle, a change in the unit cost, inclusive of an increase in a unit tax, is passed through to price on a one-to-one basis for the case of linear demand. This is not the case for a monopolist who maximises expected profit. Secondly, it seems that monopolies in the market for performance goods, such as selling access to football, price in the inelastic range of the demand curve. A similar finding applies to the monopoly market for tobacco in Sweden. For both cases it seem reasonable to say that production is characterised by a fixed cost and negligible variable cost. This is also the behaviour that is predicted when we apply the safety-first principle to a price setting monopolist. 8 Thirdly, Kahneman and Tversky (1979) argue that risk is more acceptable when gains are less likely. On this point our results show that a higher fixed cost (which makes it more likely that the profit is unacceptably low) implies that the optimum priceand, therefore, risk-goes up. Finally, the hypothesis that the monopolist uses the safety-first principle goes some way in explaining the fact that price responds more strongly to cost changes than to demand changes (cf. Kahneman et al., 1986).
Previous examinations of monopolistic pricing under uncertainty, for example Baron (1971), Leland (1972), Kimball (1989) and Hau (2004), analyse the monopolist's optimal response to uncertainty under the assumption that the firm always survives. Our discussion supplements the standard analyses of monopolistic pricing under demand uncertainty because, in contrast, we assume that the monopolistic firm uses price to minimise the risk of financial default. The safety-first principle takes into account poor performances that correspond to the tail part of a profit's distribution. This concern seems relevant unless, for example, the firm's future gains fund current poor performance (Roy, 1952). Day et al. (1971) and Arzac (1976) show that a quantity-setting monopolist sets output lower than the quantity that maximises expected profits when avoidance of ruin matters. Our result extends this because we show that the safetyfirst criterion implies a lower price when a reduction in price reduces the variance of profit, and a higher price whenever the profit's variance is a decreasing function of the price. Together, these results show that it matters whether a monopolist sets price or quantity when the aim is to minimise the probability of a financially poor performance.